The <product topology> has <basis>
$$
\boxed{\{U\times V:U\subseteq X\text{ open},\ V\subseteq Y\text{ open}\}.}
$$
<Open> subsets of the product are arbitrary unions of these rectangles. The rectangles cover the product and their finite intersections are rectangles, so they satisfy the <basis> axioms. Equivalently, this is the smallest <topology> making both coordinate projections <continuous>.

Solution

Past exam of the topology course of the University of Oxford / 2014 / 2 / a / i / Solution

 Solution (source code)